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Discrete Mathematics



Gates of convex sets

The notion of a gate for a subset in a metric space plays a role in localization theory (belonging to operations research) and in the theory of buildings (belonging to geometry). It is investigated in normed vector spaces

Theory of the incidence geometries of type L.Af* satisfying an intersection axiom

Generalizing results on the geometries with diagrams C.Af* and Af.Af*, it is possible to show that under certain hypothesis, every geometry on L.Af* is embeddable in a geometry on L.A2 and to pursue its study on this basis

The permutohedron as a permutograph

The permotuhedron is a graph used both in geometry and in the study of rankings. It is shown to belong to a whole family of graphs, called the permutographs. The automorphisms and the polytopal character of these graphs are investigated.

Arithmetic and grouptheoretical reduction of rank 3 amalgams of thick linear spaces that are flag-homogeneous

The polynomial structure of orders of thick flag-homogeneous linear spaces puts strong constraints on the orders of rank 3 amalgams of such geometries  from which a reduced list of diagrams is derived. Under the additional hypothesis where the amalgam is endowed with a chamber-transitive automorphism group the analysis of rank one parabolic subgroups leads to further reductions